Phase coexistence and general properties of phase transitions

In order to understand a little bit more about phase transitions and the coexistence of different phases, let us consider a system in the liquid-gaseous transition, and let us focus on the region of liquid-gas coexistence; physically, what happens (and what can also be understood from the figure) is that the system exhibits both a liquid and a gaseous phase, and it is possible to change the volume of the system without changing its pressure at fixed temperature. This means that the work we do on the system is used only to change the proportion between the phases (breaking or forming molecular bonds, on a microscopic level).

Let us now consider a slightly more general situation: suppose we have an isolated single-component thermodynamic system described by the internal energy and extensive variables (generalized displacements); if the system is subject to reversible processes then from the expression of we get:

where are the generalized forces relative to the generalized displacements . Let us now suppose that two phases and coexist and that they can exchange and (namely, they are systems in contact with each other). At equilibrium the total entropy must be maximized, and since the whole system is in equilibrium we also have and . Substituting in we get:
Therefore, since and are arbitrary positive quantities, we must have:

Considering our system again, since the generalized displacements that are needed in order to describe it are the volume and the number of particles , when two phases coexist we have:

Since, as we have shown in Thermodynamic potentials, the Gibbs free energy is (for a single-component system) then and so when two phases coexist we also have:
This equality must hold along the whole coexistence line in space and so if we know a point in this space where the two phases coexist we can, at least locally, "reconstruct" the coexistence line. In fact we have , namely:
where and , and by definition is the molar latent heat needed to bring the system from phase to phase . This is known as Clausius-Clapeyron equation.

From the expression of we can also understand some very general properties of phase transitions. In fact, from the expressions of and at phase coexistence we have:

This means that when the system undergoes a phase transition its volume and its entropy have a jump discontinuity; as we will later see in more detail, since in this transition the first derivatives of a thermodynamic potential have a jump discontinuity, we call it a first order transition. A similar behaviour can be encountered in magnetic systems, where the magnetization has a jump at for temperatures lower that the critical one; in this case since we see that the first derivative of the free energy with respect to has a jump discontinuity.

Furthermore, if we consider our system at the critical point we see from the projection of the phase diagram that the isothermal compressibility:

diverges when . Similarly, the magnetic susceptibility of a magnetic system at its critical temperature diverges when and . As we will see further on in Long range correlations, the divergence of response functions is a typical behaviour of thermodynamic systems in the neighbourhood of critical points and has important consequences.